# 2.6 A Sample Problem

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The length of time X, needed by students in a particular course to complete a 1 hour exam is a random variable with PDF given by



For the random variable X,

1. Find the value k that makes f(x) a probability density function (PDF)
2. Find the cumulative distribution function (CDF)
3. Graph the PDF and the CDF
4. Find the probability that that a randomly selected student will finish the exam in less than half an hour
5. Find the mean time needed to complete a 1 hour exam
6. Find the variance and standard deviation of X

## Solution

### Part 1

The given PDF must integrate to 1. Thus, we calculate



Therefore, k = 6/5. Notice also that the PDF is nonnegative everywhere.

### Part 2

The CDF, F(x), is the area function of the PDF, obtained by integrating the PDF from negative infinity to an arbitrary value x.

If x is in the interval (-∞, 0), then



If x is in the interval [0, 1], then



If x is in the interval (1, ∞) then



The CDF is therefore given by



### Part 3

The PDF and CDF of X are shown below.

### Part 4

The probability that a student will complete the exam in less than half an hour is Pr(X < 0.5). Note that since Pr(X = 0.5) = 0 (since X is a continuous random variable) it is equivalent to calculate Pr(x ≤ 0.5). This is precisely F(0.5):



### Part 5

The mean time to complete a 1 hour exam is the expected value of the random variable X. Consequently, we calculate



### Part 6

To find the variance of X, we use our alternate formula to calculate



Finally, we see that the standard deviation of X is